6.3 Bernoulli's Theorem, Pressure Types & Venturi

Bernoulli's Theorem, Pressure Types & Venturi

Moving fluids trade pressure and speed. Bernoulli’s theorem is the Module 2 statement of that trade for ideal flow, and it underpins venturi tubes, pitot-static instruments, and the pressure differences that later modules associate with lift. This section defines static, dynamic, and total pressure, states Bernoulli clearly, and applies it to a venturi — with only a light link to aerofoils and Module 8.

Pressure Types in a Flowing Fluid

Static pressure

Static pressure is the pressure exerted by the fluid on a surface due to the random molecular motion — the thermodynamic pressure of the fluid — measured by a port that does not face the flow head-on in a way that stagnates it. On an aircraft, static ports are placed where the local flow is as nearly undisturbed as practical so the instrument system samples ambient static pressure. Altimeters and vertical speed indicators use static pressure; the airspeed system uses it as one of two inputs.

In still air, static pressure is simply the atmospheric pressure at that location. In moving air, static pressure can vary from point to point along the flow (the Bernoulli effect).

Dynamic pressure

Dynamic pressure (often q) is the pressure equivalent of the kinetic energy per unit volume of the moving fluid. For ideal incompressible flow:

q = ½ ρ v²

• ρ = fluid density (kg/m³)
• v = flow speed relative to the probe or body (m/s)
• q has units of pascals (N/m²)

Dynamic pressure grows with the square of speed and linearly with density. That is why indicated airspeed and true aerodynamic loads depend strongly on how fast the aircraft moves through the air and on air density (altitude, temperature).

Total (stagnation) pressure

Total pressure — also called stagnation pressure when the flow is brought to rest isentropically — is the pressure measured by a forward-facing pitot tube that stagnates the oncoming flow. In ideal incompressible Bernoulli flow:

p_total = p_static + p_dynamic = p_static + ½ ρ v²

So:

p_dynamic = p_total − p_static

The airspeed indicator is essentially a differential pressure instrument comparing pitot (total) and static pressures; the difference is interpreted as airspeed for the calibrated scale.

Bernoulli’s Theorem

Bernoulli’s theorem (ideal form used in Module 2):

For steady, incompressible, non-viscous flow along a streamline, the sum of static pressure energy, kinetic energy per unit volume, and gravitational potential energy per unit volume remains constant:

p + ½ ρ v² + ρ g h = constant

Along a streamline where height change is negligible (many aircraft instrument and venturi problems):

p + ½ ρ v² ≈ constant

Immediate consequences:

• Where velocity increases, static pressure decreases.
• Where velocity decreases, static pressure increases.
• Total pressure p + ½ρv² stays the same along the ideal streamline (no losses).

Real fluids have viscosity and can separate; Bernoulli is an idealisation. It remains the correct conceptual tool for exam questions and for understanding why a constriction drops pressure and why a curved aerofoil surface can sustain a pressure difference.

Worked qualitative example. Air enters a duct at low speed and high static pressure, then passes a narrowed section at higher speed. At the narrow section, static pressure is lower than upstream. Downstream, if the duct widens again and flow stays attached, speed falls and static pressure recovers toward the original value (minus real losses).

Worked numeric sketch. Suppose along a streamline ρ = 1.2 kg/m³ is constant, height fixed. At point 1: v₁ = 20 m/s, p₁ = 101 000 Pa. At point 2: v₂ = 40 m/s. Then

½ρv₁² = 0.5 × 1.2 × 400 = 240 Pa
½ρv₂² = 0.5 × 1.2 × 1600 = 960 Pa

Δ(½ρv²) = 720 Pa, so p₂ ≈ p₁ − 720 Pa = 100 280 Pa if Bernoulli holds. Doubling speed quadrupled dynamic pressure (because of v²), stealing more from static pressure.

The Venturi Tube

A venturi is a duct with a smooth convergent section, a throat (minimum area), and often a divergent recovery section.

1. Continuity (incompressible): A v = constant along the duct for steady flow (volume flow rate conserved). Smaller area → higher velocity at the throat.
2. Bernoulli: higher throat velocity → lower static pressure at the throat.

Applications:

• Carburettor venturi (piston engines): low throat pressure draws fuel into the airstream for mixing.
• Vacuum generation: some systems use venturi suction for instruments or other devices when engine-driven vacuum is arranged that way.
• Flow measurement: pressure difference between inlet and throat relates to flow rate (industrial venturi meters).

The same physics as a duct constriction: speed up → static pressure down.

Design notes

Smooth contours reduce separation so the pressure drop is predictable and recovery in the diffuser is efficient. Sharp edges and damage can destroy the intended pressure field — another reason induction and sensing passages must be clean and undamaged.

Pitot-Static Link (Instrument Physics)

• Pitot tube: faces the relative wind; measures total pressure.
• Static port: measures static pressure.
• Airspeed indicator: responds to p_total − p_static ≈ ½ρv² (with calibration and compressibility corrections at higher speeds — advanced detail).
• Altimeter: uses static pressure alone against an aneroid reference.
• Blocked pitot or static ports produce characteristic instrument errors; the physical root is wrong total or static input, not “Bernoulli failing.”

Module 2 expects you to name the pressures and state that airspeed comes from the difference between pitot and static. Detailed failure cases may appear in instruments modules; the fluid principle lives here.

Aerofoil Link to Module 8 (Do Not Fully Teach Lift Here)

An aerofoil in flight has airflow that generally moves faster over the upper surface than the lower surface for a typical lifting case (plus angle-of-attack and circulation details taught in aerodynamics). By Bernoulli’s qualitative rule, higher local speed associates with lower static pressure. The pressure difference integrated over the surface contributes to lift. Real lift theory also uses Newton’s third law (downwash) and circulation; Module 8 develops aerodynamics properly.

For Module 2, stop at:

• Velocity and static pressure are linked (Bernoulli).
• A pressure difference across a surface produces a net force (pressure × area ideas from statics).
• Pitot-static and venturi are engineered applications of the same pressure–speed relationship.

Do not memorise full lift equations or stall theory here; recognise the bridge so Module 8 does not feel disconnected.

Assumptions and Limits

Bernoulli’s simple form assumes:

• Steady flow
• Incompressible fluid (good for liquids; for air, acceptable at low Mach numbers)
• Negligible viscous losses along the streamline of interest
• Flow along a streamline (not arbitrarily between unrelated points)

High-speed compressible flow, shocks, and strong turbulence need more advanced models. Hydraulics still uses Bernoulli-style energy ideas in pipeline analysis, but Module 2’s star application is air and venturi/pitot language for technicians.

Formula Recap

• Dynamic pressure: q = ½ ρ v²
• Ideal relation: p_total = p_static + ½ ρ v²
• Bernoulli (constant height): p + ½ ρ v² = constant along a streamline
• Continuity (incompressible): A₁ v₁ = A₂ v₂
• Venturi: small A → large v → low p_static at throat

Higher velocity, lower static pressure; pitot sees total, static port sees static, and their difference is the airspeed signal. That is the Module 2 core of fluid dynamics in motion.